Massachusetts Institute of TechnologyDepartment of Physics

Physics 8.962 Spring 1999

Measuring the Metric, and Curvature versusAcceleration

c1999 Edmund Bertschinger. All rights reserved.

1 Introduction

These notes show how observers can set up a coordinate system and measure the space-time geometry using clocks and lasers. The approach is similar to that of special rela-tivity, but the reader must not be misled. Spacetime diagrams with rectilinear axes donot imply flat spacetime any more than flat maps imply a flat earth.

Cartography provides an excellent starting point for understanding the metric. Ter-restrial maps always provide a scale of the sort One inch equals 1000 miles. If themap is of a sufficiently small region and is free from distortion, one scale will suffice.However, a projection of the entire sphere requires a scale that varies with location andeven direction. The Mercator projection suggests that Greenland is larger than SouthAmerica until one notices the scale difference. The simplest map projection, with lat-itude and longitude plotted as a Cartesian grid, has a scale that depends not only onposition but also on direction. Close to the poles, one degree of latitude represents a fargreater distance than one degree of longitude.

The map scale is the metric. The spacetime metric has the same meaning and use: ittranslates coordinate distances and times (one inch on the map) to physical (proper)distances and times.

The terrestrial example also helps us to understand how coordinate systems can bedefined in practice on a curved manifold. Let us consider how coordinates are defined onthe Earth. First pick one point and call it the north pole. The pole is chosen along therotation axis. Now extend a family of geodesics from the north pole, called meridiansof longitude. Label each meridian by its longitude . We choose the meridian goingthrough Greenwich, England, and call it theprime meridian, = 0. Next, we definelatitude as an affine parameter along each meridian of longitude, scaled to pi/2 at thenorth pole and decreasing linearly to pi/2 at the point where the meridians intersect

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again (the south pole). With these definitions, the proper distance between the nearbypoints with coordinates (, ) and (+d, +d) is given by ds2 = R2(d2+cos2 d2).In this way, every point on the sphere gets coordinates along with a scale which convertscoordinate intervals to proper distances.

This example seems almost trivial. However, it faithfully illustrates the conceptsinvolved in setting up a coordinate system and measuring the metric. In particular,coordinates are numbers assigned by obsevers who exchange information with each other.There is no conceptual need to have the idealized dense system of clocks and rods fillingspacetime. Observe any major civil engineering project. The metric is measured by twosurveyors with transits and tape measures or laser ranging devices. Physicists can do thesame, in principle and in practice. These notes illustrate this through a simple thoughtexperiment. The result will be a clearer understanding of the relation between curvature,gravity, and acceleration.

2 The metric in 1+1 spacetime

We study coordinate systems and the metric in the simplest nontrivial case, spacetimewith one space dimension. This analysis leaves out the issue of orientation of spatial axes.It also greatly reduces the number of degrees of freedom in the metric. As a symmetric2 matrix, the metric has three independent coefficients. Fixing two coordinates imposestwo constraints, leaving one degree of freedom in the metric. This contrasts with the sixmetric degrees of freedom in a 3+1 spacetime. However, if one understands well the 1+1example, it is straightforward (albeit more complicated) to generalize to 2+1 and 3+1spacetime.

We will construct a coordinate system starting from one observer called A. ObserverA may have any motion whatsoever relative to other objects, including acceleration.But neither spatial position nor velocity is meaningful for A before we introduce otherobservers or coordinates (velocity relative to what?) althoughAs acceleration (relativeto a local inertial frame!) is meaningful: A stands on a scale, reads the weight, anddivides by rest mass. Observer A could be you or me, standing on the surface of theearth. It could equally well be an astronaut landing on the moon. It may be helpfulin this example to think of the observers as being stationary with respect to a massivegravitating body (e.g. a black hole or neutron star). However, we are considering acompletely general case, in which the spacetime may not be at all static. (That is, theremay not be any Killing vectors whatsoever.)

We take observer As worldine to define the t-axis: A has spatial coordinate xA 0.A second observer, some finite (possibly large) distance away, is denoted B. Both A andB carry atomic clocks, lasers, mirrors and detectors.

Observer A decides to set the spacetime coordinates over all spacetime using thefollowing procedure, illustrated in Figure 1. First, the reading of As atomic clock gives

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Figure 1: Setting up a coordinate system in curved spacetime. There are two time-like worldlines and two pairs of null geodesics. The appearance of flat coordinates ismisleading; the metric varies from place to place.

the t-coordinate along the t-axis (x = 0). Then, A sends a pair of laser pulses to B, whoreflects them back to A with a mirror. If the pulses do not return with the same timeseparation (measured by A) as they were sent, A deduces that B is moving and sendssignals instructing B to adjust her velocity until t6 t5 = t2 t1. The two continuallyexchange signals to ensure that this condition is maintained. A then declares that B hasa constant space coordinate (by definition), which is set to half the round-trip light-traveltime, xB 12(t5 t1). A sends signals to inform B of her coordinate.

Having set the spatial coordinate, A now sends time signals to define the t-coordinatealong Bs worldline. As laser encodes a signal from Event 1 in Figure 1, This pulsewas sent at t = t1. Set your clock to t1 + xB. B receives the pulse at Event 3 and setsher clock. A sends a second pulse from Event 2 at t2 = t1 +t which is received by Bat Event 4. B compares the time difference quoted by A with the time elapsed on heratomic clock, the proper time B. To her surprise, B 6= t.

At first A and B are sure something went wrong; maybe B has begun to drift. Butrepeated exchange of laser pulses shows that this cannot be the explanation: the round-trip light-travel time is always the same. Next they speculate that the lasers may betraveling through a refractive medium whose index of refraction is changing with time.(A constant index of refraction wouldnt change the differential arrival time.) However,they reject this hypothesis when they find that Bs atomic clock continually runs at adifferent rate than the timing signals sent by A, while the round-trip light-travel time

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Figure 2: Testing for space curvature.

measured by A never changes. Moreover, laboratory analysis of the medium betweenthem shows no evidence for any change.

Becoming suspicious, B decides to keep two clocks, an atomic clock measuring Band another set to read the time sent by A, denoted t. The difference between the twogrows increasingly large.

The observers next speculate that they may be in a non-inertial frame so that specialrelativity remains valid despite the apparent contradiction of clock differences (gtt 6= 1)with no relative motion (dxB/dt = 0). We will return to this speculation in Section 3. Inany case, they decide to keep track of the conversion from coordinate time (sent by A)to proper time (measured by B) for nearby events on Bs worldline by defining a metriccoefficient:

gtt(t, xB) limt0

(Bt

)2. (1)

The observers now wonder whether measurements of spatial distances will yield asimilar mystery. To test this, a third observer is brought to help in Figure 2. ObserverC adjusts his velocity to be at rest relative to A. Just as for B, the definition of restis that the round-trip light-travel time measured by A is constant, t8 t1 = t9 t2 =2xC 2(xB + x). Note that the coordinate distances are expressed entirely in termsof readings of As clock. A sends timing signals to both B and C. Each of them setsone clock to read the time sent by A (corrected for the spatial coordinate distance xBand xC , respectively) and also keeps time by carrying an undisturbed atomic clock. Theformer is called coordinate time t while the latter is called proper time.

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The coordinate time synchronization provided by A ensures that t2 t1 = t5 t3 =t6 t4 = t7 t5 = t9 t8 = 2x. Note that the procedure used by A to set t and xrelates the coordinates of events on the worldlines of B and C:

(t4, x4) = (t3, x3) + (1, 1)x , (t5, x5) = (t4, x4) + (1,1)x ,(t6, x6) = (t5, x5) + (1, 1)x , (t7, x7) = (t6, x6) + (1,1)x . (2)

Because they follow simply from the synchronization provided by A, these equationsare exact; they do not require x to be small. However, by themselves they do notimply anything about the physical separations between the events. Testing this meansmeasuring the metric.

To explore the metric, C checks his proper time and confirms Bs observation thatproper time differs from coordinate time. However, the metric coefficient he deduces,gtt(xC , t), differs from Bs. (The difference is first-order in x.)

The pair now wonder whether spatial coordinate intervals are similarly skewed relativeto proper distance. They decide to measure the proper distance between them by usinglaser-ranging, the same way that A set their spatial coordinates in the first place. Bsends a laser pulse at Event 3 which is reflected at Event 4 and received back at Event5 in Figure 2. From this, she deduces the proper distance of C,

s =1

2(5 3) (3)

where i is the reading of her atomic clock at event i. To her surprise, B finds that xdoes not measure proper distance, not even in the limit x 0. She defines anothermetric coefficient to convert coordinate distance to proper distance,

gxx limx0

(s

x

)2. (4)

The measurement of proper distance in equation (4) must be made at fixed t; oth-erwise the distance must be corrected for relative motion between B and C (shouldany exist). Fortunately, B can make this measurement at t = t4 because that is whenher laser pulse reaches C (see Fig. 2 and eqs. 2). Expanding 5 = B(t4 + x) and3 = B(t4 x) to first order in x using equations (1), (3), and (4), she finds

gxx(x, t) = gtt(x, t) . (5)The observers repeat the experiment using Events 5, 6, and 7. They find that, while themetric may have changed, equation (5) still holds.

The observers are intrigued to find such a relation between the time and space partsof their metric, and they wonder whether this is a general phenomenon. Have theydiscovered a modification of special relativity, in which the Minkowski metric is simplymultipled by a conformal factor, g =

2?

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They decide to explore this question by measuring gtx. A little thought shows thatthey cannot do this using pairs of events with either fixed x or fixed t. Fortunately, theyhave ideal pairs of events in the lightlike intervals between Events 3 and 4:

ds234 limt,x0

gtt(t4 t3)2 + 2gtx(t4 t3)(x4 x3) + gxx(x4 x3)2 . (6)

Using equations (2) and (5) and the condition ds = 0 for a light ray, they conclude

gtx = 0 . (7)

Their space and time coordinates are orthogonal but on account of equations (5) and (7)all time and space intervals are stretched by

gxx.

Our observers now begin to wonder if they have discovered a modification of specialrelativity, or perhaps they are seeing special relativity in a non-inertial frame. However,we know better. Unless the Riemann tensor vanishes identically, the metric they havedetermined cannot be transformed everywhere to the Minkowski form. Instead, whatthey have found is simply a consequence of how A fixed the coordinates. Fixing twocoordinates means imposing two gauge conditions on the metric. A defined coordinatesso as to make the problem look as much as possible like special relativity (eqs. 2).Equations (5) and (7) are the corresponding gauge conditions.

It is a special feature of 1+1 spacetime that the metric can always be reduced to aconformally flat one, i.e.

ds2 = 2(x)dxdx (8)

for some function (x) called the conformal factor. In two dimensions the Riemanntensor has only one independent component and the Weyl tensor vanishes identically.Advanced GR and differential geometry texts show that spacetimes with vanishing Weyltensor are conformally flat.

Thus, A has simply managed to assign conformally flat coordinates. This isnt acoincidence; by defining coordinate times and distances using null geodesics, he forcedthe metric to be identical to Minkowski up to an overall factor that has no effect on nulllines. Equivalently, in two dimensions the metric has one physical degree of freedom,which has been reduced to the conformal factor gxx = gtt.

This does not mean that A would have had such luck in more than two dimensions.In n dimensions the Riemann tensor has n2(n2 1)/12 independent components (Waldp. 54) and for n 3 the Ricci tensor has n(n+1)/2 independent components. For n = 2and n = 3 the Weyl tensor vanishes identically and spacetime is conformally flat. Notso for n > 3.

It would take a lot of effort to describe a complete synchronization in 3+1 spacetimeusing clocks and lasers. However, even without doing this we can be confident thatthe metric will not be conformally flat except for special spacetimes for which the Weyltensor vanishes. Incidentally, in the weak-field limit conformally flat spacetimes have

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no deflection of light (can you explain why?). The solar deflection of light rules outconformally flat spacetime theories including ones proposed by Nordstrom and Weyl.

It is an interesting exercise to show how to transform an arbitrary metric of a 1+1spacetime to the conformally flat form. The simplest way is to compute the Ricci scalar.For the metric of equation (8), one finds

R = 2(2t 2x) ln2 . (9)

Starting from a 1+1 metric in a different form, one can compute R everywhere in space-time. Equation (9) is then a nonlinear wave equation for (t, x) with source R(t, x). Itcan be solved subject to initial Cauchy data on a spacelike hypersurface on which = 1,t = x = 0 (corresponding to locally flat coordinates).

We have exhausted the analysis of 1+1 spacetime. Our observers have discerned onepossible contradiction with special relativity: clocks run at different rates in differentplaces (and perhaps at different times). If equation (9) gives Ricci scalar R = 0 ev-erywhere with =

gtt, then the spacetime is really flat and we must be seeing theeffects of accelerated motion in special relativity. If R 6= 0, then the variation of clocksis an entirely new phenomenon, which we call gravitational redshift.

3 The metric for an accelerated observer

It is informative to examine the problem from another perspective by working out themetric that an arbitrarily accelerating observer in a flat spacetime would deduce usingthe synchronization procedure of Section 2. We can then more clearly distinguish theeffects of curvature (gravity) and acceleration.

Figure 3 shows the situation prevailing in special relativity when observer A hasan arbitrary timelike trajectory xA(A) where A is the proper time measured by hisatomic clock. While As worldline is erratic, those of light signals are not, because heret = x0 and x = x1 are flat coordinates in Minkowski spacetime. Given an arbitraryworldline xA(A), how can we possibly find the worldines of observers at fixed coordinatedisplacement as in the preceding section?

The answer is the same as the answer to practically all questions of measurement inGR: use the metric! The metric of flat spacetime is the Minkowski metric, so the paths oflaser pulses are very simple. We simply solve an algebra problem enforcing that Events1 and 2 are separated by a null geodesic (a straight line in Minkowski spacetime) andlikewise for Events 2 and 3, as shown in Figure 3. Notice that the lengths (i.e. coordinatedifferences) of the two null rays need not be the same.

The coordinates of Events 1 and 3 are simply the coordinates along As worldine,while those for Event 2 are to be determined in terms of As coordinates. As in Section 2,A defines the spatial coordinate of B to be twice the round-trip light-travel time. Thus,if event 0 has x0 = tA(0), then Event 3 has x

0 = tA(0 + 2L). For convenience we will

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Figure 3: An accelerating observer sets up a coordinate system with an atomic clock,laser and detector.

set 0 AL. Then, according to the prescription of Section 2, A will assign to Event2 the coordinates (A, L). The coordinates in our flat Minkowksi spacetime are

Event 1: x0 = tA(A L) , x1 = xA(A L) ,Event 2: x0 = t(A, L) , x

1 = x(A, L) ,

Event 3: x0 = tA(A + L) , x1 = xA(A + L) . (10)

Note that the argument A for Event 2 is not an affine parameter along Bs wordline;it is the clock time sent to B by A. A second argument L is given so that we can lookat a family of worldlines with different L. A is setting up coordinates by finding thespacetime paths corresponding to the coordinate lines L = constant and A = constant.We are performing a coordinate transformation from (t, x) to (A, L).

Requiring that Events 1 and 2 be joined by a null geodesic in flat spacetime givesthe condition x2 x1 = (C1, C1) for some constant C1. The same condition for Events2 and 3 gives x3 x2 = (C2,C2) (with a minus sign because the light ray travelstoward decreasing x). These conditions give four equations for the four unknowns C1,C2, t(A, L), and x(A, L). Solving them gives the coordinate transformation between(A, L) and the Minkowski coordinates:

t(A, L) =1

2[tA(A + L) + tA(A L) + xA(A + L) xA(A L)] ,

x(A, L) =1

2[xA(A + L) + xA(A L) + tA(A + L) tA(A L)] . (11)

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Note that these results are exact; they do not assume that L is small nor do they restrictAs worldline in any way except that it must be timelike. The student may easily evaluateC1 and C2 and show that they are not equal unless xA(A + L) = xA(A L).

Using equations (11), we may transform the Minkowski metric to get the metric inthe coordinates A has set with his clock and laser, (A, L):

ds2 = dt2 + dx2 = gttd 2A + 2gtxdAdL+ gxxdL2 . (12)Substituting equations (11) gives the metric components in terms of As four-velocitycomponents,

gtt = gxx =[V tA(A + L) + V

xA (A + L)

] [V tA(A L) V xA (A L)

], gtx = 0 . (13)

This is precisely in the form of equation (8), as it must be because of the way in whichA coordinatized spacetime.

It is straightforward to work out the Riemann tensor from equation (13). Not surpris-ingly, it vanishes identically. Thus, an observer can tell, through measurements, whetherhe or she lives in a flat or nonflat spacetime. The metric is measurable.

Now that we have a general result, it is worth simplifying to the case of an observerwith constant acceleration gA in Minkowski spacetime. Problem 3 of Problem Set 1showed that one can write the trajectory of such an observer (up to the addition ofconstants) as x = g1A cosh gAA, t = g

1A sinh gAA. Equation (13) then gives

ds2 = e2gAL(d 2A + dL2) . (14)

One word of caution is in order about the interpretation of equation (14). Ourderivation assumed that the acceleration gA is constant for observer A at L = 0. However,this does not mean that other observers (at fixed, nonzero L) have the same acceleration.To see this, we can differentiate equations (11) to derive the 4-velocity of observer B at(A, L) and the relation between coordinate time A and proper time along Bs worldline,with the result

V B (A, L) = (cosh gAA, sinh gAA) = (cosh gBB, sinh gBB) ,dBdA

=gAgB

= egL . (15)

The four-acceleration of B follows from aB = dVB /dB = e

gLdV /dA and its mag-nitude is therefore gB = gAe

gL. The proper acceleration varies with L precisely sothat the proper distance between observers A and B, measured at constant A, remainsconstant.

4 Gravity versus acceleration in 1+1 spacetime

Equation (14) gives one form of the metric for a flat spacetime as seen by an acceleratingobserver. There are many other forms, and it is worth noting some of them in order to

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gain some intuition about the effects of acceleration. For simplicity, we will restrict ourdiscussion here to static spacetimes, i.e. metrics with g0i = 0 and tg = 0. In 1+1spacetime this means the line element may be written

ds2 = e2(x)dt2 + e2(x)dx2 . (16)

(The metric may be further transformed to the conformally flat form, eq. 8, but we leaveit in this form because of its similarity to the form often used in 3 + 1 spacetime.)

Given the metric (16), we would like to know when the spacetime is flat. If it is flat,we would like the explicit coordinate transformation to Minkowski. Both of these arestraightforward in 1+1 spacetime. (One might hope for them also to be straightforwardin more dimensions, at least in principle, but the algebra rapidly increases.)

The definitive test for flatness is given by the Riemann tensor. Because the Weyltensor vanishes in 1+1 spacetime, it is enough to examine the Ricci tensor. With equation(16), the Ricci tensor has nonvanishing components

Rtt = e+ dg

dx, Rxx = e(+) dg

dxwhere g(x) = e g(x) = e+

d

dx. (17)

The function g(x) is the proper acceleration along the x-coordinate line, along whichthe tangent vector (4-velocity) is V x = e

(1, 0). This follows from computing the4-acceleration with equation (16) using the covariant prescription a(x) = V V =V x V x . The magnitude of the acceleration is then g(x) (gaa)1/2, yielding g(x) =ed/dx. The factor e converts d/dx to g(x) = d/dl where dl =

gxx dx measures

proper distance.A stationary observer, i.e. one who remains at fixed spatial coordinate x, feels a time-

independent effective gravity g(x). Nongravitational forces (e.g. a rocket, or the contactforce from a surface holding the observer up) are required to maintain the observer atfixed x. The gravity field g(x) can be measured very simply by releasing a test particlefrom rest and measuring its acceleration relative to the stationary observer. For example,we measure g on the Earth by dropping masses and measuring their acceleration in thelab frame.

We will see following equation (18) below why the function g(x) = (d/dt)g(x) ratherthan g(x) determines curvature. For now, we simply note that equation (17) implies thatspacetime curvature is given (for a static 1+1 metric) by the gradient of the gravitationalredshift factor

gtt = e rather than by the gravity (i.e. acceleration) gradientdg/dx.

In linearized gravitation, g = g and so we deduced (in the notes Gravitation inthe Weak-Field Limit) that a spatially uniform gravitational (gravitoelectric) field canbe transformed away by making a global coordinate transformation to an acceleratingframe. For strong fields, g 6= g and it is no longer true that a uniform gravitoelectric fieldcan be transformed away. Only if the gravitational redshift factor e(x) varies linearly

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with proper distance, i.e. g d(e)/dl is a constant, is spacetime is flat, enabling oneto transform coordinates so as to remove all evidence for acceleration. If, on the otherhand, dg/dx 6= 0 even if dg/dx = 0 then the spacetime is not flat and no coordinatetransformation can transform the metric to the Minkowski form.

Suppose we have a line element for which g(x) = constant. We know that such aspacetime is flat, because the Ricci tensor (hence Riemann tensor, in 1+1 spacetime)vanishes everywhere. What is the coordinate transformation to Minkowski?

The transformation may be found by writing the metric as g = T where =x/x is the Jacobian matrix for the transformation x(x). (Note that here g is thematrix with entries g and not the gravitational acceleration!) By writing t = t(t, x)and x = x(t, x), substituting into g = T, using equation (16) and imposing theintegrability conditions 2t/tx = 2t/xt and 2x/tx = 2x/xt, one finds

t(t, x) =1

gsinh g t , x(t, x) =

1

gcosh g t if

dg

dx= 0 , (18)

up to the addition of irrelevant constants. We recognize this result as the trajectory inflat spacetime of a constantly accelerating observer.

Equation (18) is easy to understand in light of the discussion following equation (14).The proper time for the stationary observer at x is related to coordinate time t byd =

gtt(x) dt = edt. Thus, g(x) = e g t = g t or, in the notation of equation(15), gBB = gAA (since A was used there as the global t-coordinate). The conditione g = g(x) = constant amounts to requiring that all observers be able to scale theirgravitational accelerations to a common value for the observer at (x) = 0, g. If theycannot (i.e. if dg/dx 6= 0), then the metric is not equivalent to Minkowski spacetimeseen in the eyes of an accelerating observer.

With equations (16)(18) in hand, we can write the metric of a flat spacetime inseveral new ways, with various spatial dependence for the acceleration of our coordinateobservers:

ds2 = e2g x(dt2 + dx2) , g(x) = g eg x (19)= g2(x x0)2dt2 + dx2 , g(x) = 1

x x0 (20)

= [2g(x x0)]dt2 + [2g(x x0)]1dx2 , g(x) =

g

2(x x0) . (21)

The first form was already given above in equation (14). The second and third forms arepeculiar in that there is a coordinate singularity at x = x0; these coordinates only workfor x > x0. This singularity is very similar to the one occuring in the Schwarzschild lineelement. Using the experience we have obtained here, we will remove the Schwarzschildsingularity at r = 2GM by performing a coordinate transformation similar to those used

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here. The student may find it instructive to write down the coordinate transformationsfor these cases using equation (18) and drawing the (t, x) coordinate lines on top ofthe Minkowski coordinates (t, x). While the singularity at x = x0 can be transformedaway, it does signal the existence of an event horizon. Equation (20) is called Rindlerspacetime. Its event horizon is discussed briefly in Schutz (p. 150) and in more detailby Wald (pp. 149152).

Actually, equation (21) is closer to the Schwarzschild line element. Indeed, it becomesthe r-t part of the Schwarzschild line element with the substitutions x r, 2gx0 1and g GM/r2. These identifications show that the Schwarzschild spacetime differsfrom Minkowski in that the acceleration needed to remain stationary is radially directedand falls off as e r2. We can understand many of its features through this identificationof gravity and acceleration.

For completeness, I list three more useful forms for a flat spacetime line element:

ds2 = dt2 + g2(t t0)2dx2 , g(x) = 0 (22)= dUdV (23)= evududv . (24)

The first is similar to Rindler spacetime but with t and x exchanged. The result issuprising at first: the acceleration of a stationary observer vanishes. Equation (22) hasthe form of Gaussian normal or synchronous coordinates (Wald, p. 42). It representsthe coordinate frame of a freely-falling observer. It is interesting to ask why, if theobserver is freely-falling, the line element does not reduce to Minkowski despite the factthat this spacetime is flat. The answer is that different observers (i.e., worldlines ofdifferent x) are in uniform motion relative to one another. In other words, equation (22)is Minkowski spacetime in expanding coordinates. It is very similar to the Robertson-Walker spacetime, which reduces to it (short of two spatial dimensions) when the massdensity is much less than the critical density.

Equations (23) and (24) are Minkowski spacetime in null (or light-cone) coordinates.For example, U = t x, V = t+ x. These coordinates are useful for studying horizons.

Having derived many results in 1 + 1 spacetime, I close with the cautionary remarkthat in 2 + 1 and 3 + 1 spacetime, there are additional degrees of freedom in the met-ric that are quite unlike Newtonian gravity and cannot be removed (even locally) bytransformation to a linearly accelerating frame. Nonetheless, it should be possible toextend the treatment of these notes to account for these effects gravitomagnetism andgravitational radiation. As shown in the notes Gravitation in the Weak-Field Limit, auniform gravitomagnetic field is equivalent to uniformly rotating coordinates. Gravita-tional radiation is different; there is no such thing as a spatially uniform gravitationalwave. However, one can always choose coordinates so that gravitational radiation strainsij and its first derivatives vanish at a point.

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